Covers of Finitely Generated Acts over Monoids

In (Semigroup Forum 77: 325-338, 2008) Mahmoudi M. and Renshaw J. solved a study that covers of cyclic S-acts over monoids. This article is an attempt to initiate the covers of finitely generated S-acts. We give a necessary and sufficient condition for a monoid to have the properties that n-generated S-acts have strongly flat covers, Condition $(P)$ covers and projective covers. The main conclusions extend some known results. We show also that Condition $(P)$ covers of finitely generated S-acts are not unique, unlike the situation for strongly flat covers. Additionally, we demonstrate that the property of Enochs' X-precover of S-act A, where X denotes a class of S-acts that are closed under isomorphisms.